Group #42130433 — subgroups

Subgroups — C₁₀₉×#12439 #42130433

Order 27904.

248 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2246	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#1668	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1371	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₆×C₁₀₉	—	2	1	no	—
7	C₂×D₁₆×C₁₀₉	—	2	1	no	—
8	#493×C₁₀₉	C₂²	1	2	yes	—
9	#460×C₁₀₉	—	2	2	no	—
10	C₈×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₁₆×C₁₀₉	—	2	1	no	—
12	C₄⋊C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₁₀₉	—	2	1	no	—
14	C₄×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	#335×C₁₀₉	C₂²	1	2	yes	—
16	#331×C₁₀₉	—	2	2	no	—
17	C₂×SD₁₆×C₁₀₉	—	4	2	no	—
18	C₂×D₈×C₁₀₉	—	2	2	no	—
19	C₂×D₈×C₁₀₉	—	2	2	no	—
20	#127×C₁₀₉	—	2	2	no	—
21	C₄×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
22	D₁₆×C₁₀₉	—	4	2	no	—
23	D₁₆×C₁₀₉	—	4	2	no	—
24	C₂×C₁₆×C₁₀₉	—	2	1	no	—
25	C₂×C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₁₀₉	—	2	1	no	—
28	C₄⋊C₈×C₁₀₉	—	2	2	no	—
29	C₄⋊C₈×C₁₀₉	D₄	1	1	yes	characteristic
30	#102×C₁₀₉	—	4	2	no	—
31	C₄×C₈×C₁₀₉	—	2	1	no	—
32	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₀₉	D₈	1	1	yes	characteristic
34	C₂×D₄×C₁₀₉	—	4	2	no	—
35	C₂×D₄×C₁₀₉	—	4	2	no	—
36	C₂×D₄×C₁₀₉	—	4	2	no	—
37	SD₁₆×C₁₀₉	—	8	4	no	—
38	D₈×C₁₀₉	—	4	2	no	—
39	D₈×C₁₀₉	—	4	2	no	—
40	D₈×C₁₀₉	—	8	2	no	—
41	C₂×C₈×C₁₀₉	—	2	1	no	—
42	C₂×C₈×C₁₀₉	—	4	2	no	—
43	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₁₀₉	—	2	1	no	—
46	C₄⋊C₄×C₁₀₉	D₈	1	1	yes	characteristic
47	C₄²×C₁₀₉	—	2	1	no	—
48	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₁₀₉	—	2	1	no	—
50	C₁₆×C₁₀₉	—	4	1	no	—
51	C₁₆×C₁₀₉	SD₁₆	1	2	yes	complemented
52	C₂³×C₁₀₉	—	8	2	no	—
53	Q₈×C₁₀₉	—	2	1	no	—
54	Q₈×C₁₀₉	D₁₆	1	2	yes	complemented
55	D₄×C₁₀₉	—	8	2	no	—
56	D₄×C₁₀₉	—	8	2	no	—
57	D₄×C₁₀₉	—	8	2	no	—
58	D₄×C₁₀₉	—	8	2	no	—
59	D₄×C₁₀₉	—	16	2	no	—
60	C₂×C₄×C₁₀₉	—	2	1	no	—
61	C₂×C₄×C₁₀₉	—	2	1	no	—
62	C₂×C₄×C₁₀₉	#121	1	1	yes	characteristic
63	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₁₀₉	#135	1	1	yes	characteristic
65	C₈×C₁₀₉	—	2	1	no	—
66	C₈×C₁₀₉	—	4	1	no	—
67	C₈×C₁₀₉	—	8	2	no	—
68	C₈×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₈×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂²×C₁₀₉	—	16	2	no	—
71	C₂²×C₁₀₉	—	16	2	no	—
72	C₂²×C₁₀₉	—	16	2	no	—
73	C₂²×C₁₀₉	#466	1	1	yes	characteristic
74	C₄×C₁₀₉	—	2	2	no	—
75	C₄×C₁₀₉	—	2	1	no	—
76	C₄×C₁₀₉	—	4	1	no	—
77	C₄×C₁₀₉	#460	1	1	yes	characteristic
78	C₄×C₁₀₉	#472	1	1	yes	characteristic
79	C₄×C₁₀₉	C₂×D₁₆	1	1	yes	characteristic
80	C₄×C₁₀₉	#508	1	1	yes	characteristic
81	#12439	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₁₀₉	—	32	2	no	—
83	C₂×C₁₀₉	#1668	1	1	yes	characteristic
84	C₂×C₁₀₉	#2215	1	1	yes	characteristic
85	C₂×C₁₀₉	#2221	1	1	yes	characteristic
86	#2246	C₂×C₁₀₉	1	1	yes	characteristic
87	Q₈×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
88	#1668	C₂×C₁₀₉	1	2	yes	complemented
89	#1371	C₂×C₁₀₉	1	1	yes	characteristic, complemented
90	#1361	C₂×C₁₀₉	1	2	yes	complemented
91	C₁₀₉	#12439	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×D₁₆	—	2	1	no	—
93	C₂×D₁₆	—	2	1	no	—
94	#493	C₂²×C₁₀₉	1	2	yes	—
95	#460	—	2	2	no	—
96	C₈×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
97	C₄⋊C₁₆	—	2	1	no	—
98	C₄⋊C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
99	C₄×C₁₆	—	2	1	no	—
100	C₄×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
101	#335	C₂²×C₁₀₉	1	2	yes	—
102	#331	—	2	2	no	—
103	C₂×SD₁₆	—	4	2	no	—
104	C₂×D₈	—	2	2	no	—
105	C₂×D₈	—	2	2	no	—
106	#127	—	2	2	no	—
107	C₄×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
108	D₁₆	—	4	2	no	—
109	D₁₆	—	4	2	no	—
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	D₄×C₁₀₉	1	1	yes	characteristic
112	C₂×C₁₆	D₄×C₁₀₉	1	1	yes	characteristic
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	D₄×C₁₀₉	1	1	yes	characteristic
116	#102	—	4	2	no	—
117	C₄×C₈	—	2	1	no	—
118	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
119	C₂×Q₈	D₈×C₁₀₉	1	1	yes	characteristic
120	C₂×D₄	—	4	2	no	—
121	C₂×D₄	—	4	2	no	—
122	C₂×D₄	—	4	2	no	—
123	SD₁₆	—	8	4	no	—
124	D₈	—	4	2	no	—
125	D₈	—	4	2	no	—
126	D₈	—	8	2	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
130	C₂×C₈	#41×C₁₀₉	1	1	yes	characteristic
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	D₈×C₁₀₉	1	1	yes	characteristic
133	C₄²	—	2	1	no	—
134	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	—	4	1	no	—
137	C₁₆	SD₁₆×C₁₀₉	1	2	yes	complemented
138	C₂³	—	8	2	no	—
139	Q₈	—	2	1	no	—
140	Q₈	D₁₆×C₁₀₉	1	2	yes	complemented
141	D₄	—	8	2	no	—
142	D₄	—	8	2	no	—
143	D₄	—	8	2	no	—
144	D₄	—	8	2	no	—
145	D₄	—	16	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	#121×C₁₀₉	1	1	yes	characteristic
149	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
150	C₂×C₄	#135×C₁₀₉	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	4	1	no	—
153	C₈	—	8	2	no	—
154	C₈	C₂×SD₁₆×C₁₀₉	1	1	yes	characteristic
155	C₈	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
156	C₂²	—	16	2	no	—
157	C₂²	—	16	2	no	—
158	C₂²	—	16	2	no	—
159	C₂²	#466×C₁₀₉	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	1	no	—
162	C₄	—	4	1	no	—
163	C₄	#460×C₁₀₉	1	1	yes	characteristic
164	C₄	#472×C₁₀₉	1	1	yes	characteristic
165	C₄	C₂×D₁₆×C₁₀₉	1	1	yes	characteristic
166	C₄	#508×C₁₀₉	1	1	yes	characteristic
167	C₂	—	32	2	no	—
168	C₂	C₁₀₉×#1668	1	1	yes	characteristic
169	C₂	C₁₀₉×#2215	1	1	yes	characteristic
170	C₂	C₁₀₉×#2221	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2246

C₂

yes

maximal, characteristic

Q₈×C₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1668

C₂

yes

maximal, complemented

C₁₀₉×#1371

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1361

C₂

yes

maximal, complemented

C₂×D₁₆×C₁₀₉

—

C₂×D₁₆×C₁₀₉

—

#493×C₁₀₉

C₂²

yes

—

#460×C₁₀₉

—

C₈×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₀₉

—

C₄⋊C₁₆×C₁₀₉

C₂²

yes

characteristic

C₄×C₁₆×C₁₀₉

—

C₄×C₁₆×C₁₀₉

C₂²

yes

characteristic

#335×C₁₀₉

C₂²

yes

—

#331×C₁₀₉

—

C₂×SD₁₆×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

#127×C₁₀₉

—

C₄×Q₈×C₁₀₉

D₄

yes

characteristic

D₁₆×C₁₀₉

—

D₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

D₄

yes

characteristic

C₂×C₁₆×C₁₀₉

D₄

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

D₄

yes

characteristic

#102×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

D₈

yes

characteristic

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

SD₁₆×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₉

#41

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₈

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

SD₁₆

yes

complemented

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

D₁₆

yes

complemented

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#121

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₉

#135

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₈×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#466

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

C₂×D₁₆

yes

C₄×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

characteristic

Q₈×C₁₆

C₂×C₁₀₉

yes

characteristic, complemented

#1668

C₂×C₁₀₉

yes

complemented

#1371

C₂×C₁₀₉

yes

characteristic, complemented

#1361

C₂×C₁₀₉

yes

complemented

C₁₀₉

#12439

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

—

C₂×D₁₆

—

#493

C₂²×C₁₀₉

yes

—

#460

—

C₈×Q₈

C₂²×C₁₀₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²×C₁₀₉

yes

characteristic

C₄×C₁₆

—

100

C₄×C₁₆

C₂²×C₁₀₉

yes

characteristic

101

#335

C₂²×C₁₀₉

yes

—

102

#331

—

103

C₂×SD₁₆

—

104

C₂×D₈

—

105

C₂×D₈

—

106

#127

—

107

C₄×Q₈

D₄×C₁₀₉

yes

characteristic

108

D₁₆

—

109

D₁₆

—

110

C₂×C₁₆

—

111

C₂×C₁₆

D₄×C₁₀₉

yes

characteristic

112

C₂×C₁₆

D₄×C₁₀₉

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₄⋊C₈

D₄×C₁₀₉

yes

characteristic

116

#102

—

117

C₄×C₈

—

118

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

119

C₂×Q₈

D₈×C₁₀₉

yes

characteristic

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

SD₁₆

—

124

D₈

—

125

D₈

—

126

D₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂×D₄×C₁₀₉

yes

characteristic

130

C₂×C₈

#41×C₁₀₉

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

D₈×C₁₀₉

yes

characteristic

133

C₄²

—

134

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

135

C₁₆

—

136

C₁₆

—

137

C₁₆

SD₁₆×C₁₀₉

yes

complemented

138

C₂³

—

139

Q₈

—

140

Q₈

D₁₆×C₁₀₉

yes

complemented

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#121×C₁₀₉

yes

characteristic

149

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

150

C₂×C₄

#135×C₁₀₉

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

C₂×SD₁₆×C₁₀₉

yes

characteristic

155

C₈

AGL(1,ℤ/8)×C₁₀₉

yes

characteristic

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#466×C₁₀₉

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#460×C₁₀₉

yes

characteristic

164

C₄

#472×C₁₀₉

yes

characteristic

165

C₄

C₂×D₁₆×C₁₀₉

yes

characteristic

166

C₄

#508×C₁₀₉

yes

characteristic

167

C₂

—

168

C₂

C₁₀₉×#1668

yes

characteristic

169

C₂

C₁₀₉×#2215

yes

characteristic

170

C₂

C₁₀₉×#2221

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
prime radical	C₁₀₉×#2246	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈×Q₈×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₀₉

prime radical

C₁₀₉×#2246

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈×Q₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. Q₈×C₁₆×C₁₀₉	C₂	semidirect
3. C₁₀₉×#1668	C₂	semidirect
4. C₁₀₉×#1371	C₂	semidirect
5. C₁₀₉×#1361	C₂	semidirect
51. C₁₆×C₁₀₉	SD₁₆	semidirect
54. Q₈×C₁₀₉	D₁₆	semidirect
81. #12439	C₁₀₉	direct (N × H)
87. Q₈×C₁₆	C₂×C₁₀₉	semidirect
88. #1668	C₂×C₁₀₉	semidirect
89. #1371	C₂×C₁₀₉	semidirect
90. #1361	C₂×C₁₀₉	semidirect
91. C₁₀₉	#12439	direct (N × H)
137. C₁₆	SD₁₆×C₁₀₉	semidirect
140. Q₈	D₁₆×C₁₀₉	semidirect

H ≅ G/N

type

2. Q₈×C₁₆×C₁₀₉

C₂

semidirect

3. C₁₀₉×#1668

C₂

semidirect

4. C₁₀₉×#1371

C₂

semidirect

5. C₁₀₉×#1361

C₂

semidirect

51. C₁₆×C₁₀₉

SD₁₆

semidirect

54. Q₈×C₁₀₉

D₁₆

semidirect

81. #12439

C₁₀₉

direct (N × H)

87. Q₈×C₁₆

C₂×C₁₀₉

semidirect

88. #1668

C₂×C₁₀₉

semidirect

89. #1371

C₂×C₁₀₉

semidirect

90. #1361

C₂×C₁₀₉

semidirect

91. C₁₀₉

#12439

direct (N × H)

137. C₁₆

SD₁₆×C₁₀₉

semidirect

140. Q₈

D₁₆×C₁₀₉

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₀₉×#2246	2
Q₈×C₁₆×C₁₀₉	2
C₁₀₉×#1668	2
C₁₀₉×#1371	2
C₁₀₉×#1361	2
#12439	109

subgroup

index

C₁₀₉×#2246

Q₈×C₁₆×C₁₀₉

C₁₀₉×#1668

C₁₀₉×#1371

C₁₀₉×#1361

#12439

109

Subgroups — C₁₀₉×#12439 #42130433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12439 #42130433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12439 #42130433